Question

Water in a lake is changing into ice at $0^{o}C$ when the atmospheric temperature is ${10}^{o}C$. If the time taken for $1$ cm thick ice layer to be formed is $7$ hours, the time required for the thickness of ice to increase from $1$ cm to $2$ cm is

• A. 7 hours
• B. 14 hours
• C. < 7 hours
• D. > 14 hours

Answer 1

The correct option is D > 14 hours

Let x be the thickness of the ice layer at a certain time. If the thickness is increased by dx in a certain time dt, the amount of heat flowing through the slab in time dt is given by

$Q=\frac{kATdt}{dx}$ ....(1)

where A is the area of the layer of ice and -T is the temp of the surrounding air.

If dm is the mass of water frozen into ice then,

$Q=dm\ast L$

But, $dm=A\rho dx$ where $\rho$ is the density of ice.

Hence, $Q=A\rho Ldx$ ....(2)

Equating (1) and (2) we get,

$dt=\frac{\rho L}{kT}xdx$

$\Rightarrow {\int }_0^{t}dt=\frac{\rho L}{kT}{\int }_{x_1}^{x_2}xdx$

$\Rightarrow t=\frac{\rho L}{kT}(x_2^2-x_1^2)$

$t_1=\frac{\rho L}{kT}(1^2-0^2)$

$t_2=\frac{\rho L}{kT}(2^2-1^2)$

$\therefore \frac{t_2}{t_1}=\frac{4-1}{1-0}$

$\Rightarrow t_2=3t_1=3\ast 7=21hrs$

$\therefore t_2>14hrs$